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t follow his tables of refractions； making the refractions of the sun and moon （altogether against the nature of light） to exceed the refractions of the fixed stars； and that by four or five minutes NEAR THE HORIZON； did thereby increase the moon's HORIZONTAL parallax by a like number of minutes； that is； by a twelfth or fifteenth part of the whole parallax。 Correct this error and the distance will become about 60 1/2 semi…diameters of the earth； near to what others have assigned。 Let us assume the mean distance of 60 diameters in the syzygies； and suppose one revolution of the moon； in respect to the fixed stars； to be completed in 27d。 7h。 43'； as astronomers have determined； and the circumference of the earth to amount to 123；249；600 Paris feet； as the French have found by mensuration。 And now； if we imagine the moon； deprived of all motion； to be let go； so as to descend towards the earth with the impulse of all that force by which （by Cor。 Prop。 iii。） it is retained in its orb； it will in the space of one minute of time describe in its fall 15 1/12 Paris feet。 For the versed sine of that arc which the moon； in the space of one minute of time； would by its mean motion describe at the distance of sixty semi…diameters of the earth； is nearly 15 1/12 Paris feet； or more accurately 15 feet； 1 inch； 1 line 4/9。 Wherefore； since that force； in approaching the earth； increases in the reciprocal…duplicate proportion of the distance； and upon that account； at the surface of the earth； is 60 x 60 times greater than at the moon； a body in our regions； falling with that force； ought in the space of one minute of time to describe 60 x 60 x 15 1/12 Paris feet； and in the space of one second of time； to describe 15 1/12 of those feet； or more accurately； 15 feet； 1 inch； 1 line 4/9。 And with this very force we actually find that bodies here upon earth do really descend； for a pendulum oscillating seconds in the latitude of Paris will be 3 Paris feet； and 8 lines 1/2 in length； as Mr。 Huygens has observed。 And the space which a heavy body describes by falling in one second of time is to half the length of the pendulum in the duplicate ratio of the circumference of a circle to its diameter （as Mr。 Huygens has also shown）； and is therefore 15 Paris feet； 1 inch； 1 line 4/9。 And therefore the force by which the moon is retained in its orbit is that very same force which we commonly call gravity； for； were gravity another force different from that； then bodies descending to the earth with the joint impulse of both forces would fall with a double velocity； and in the space of one second of time would describe 30 1/6 Paris feet； altogether against experience。〃＇1＇ All this is beautifully clear； and its validity has never in recent generations been called in question； yet it should be explained that the argument does not amount to an actually indisputable demonstration。 It is at least possible that the coincidence between the observed and computed motion of the moon may be a mere coincidence and nothing more。 This probability； however； is so remote that Newton is fully justified in disregarding it； and； as has been said； all subsequent generations have accepted the computation as demonstrative。 Let us produce now Newton's further computations as to the other planetary bodies； passing on to his final conclusion that gravity is a universal force。 〃PROPOSITION V。； THEOREM V。 〃That the circumjovial planets gravitate towards Jupiter； the circumsaturnal towards Saturn； the circumsolar towards the sun； and by the forces of their gravity are drawn off from rectilinear motions； and retained in curvilinear orbits。
〃For the revolutions of the circumjovial planets about Jupiter； of the circumsaturnal about Saturn； and of Mercury and Venus and the other circumsolar planets about the sun； are appearances of the same sort with the revolution of the moon about the earth； and therefore； by Rule ii。； must be owing to the same sort of causes； especially since it has been demonstrated that the forces upon which those revolutions depend tend to the centres of Jupiter； of Saturn； and of the sun； and that those forces； in receding from Jupiter； from Saturn； and from the sun； decrease in the same proportion； and according to the same law； as the force of gravity does in receding from the earth。 〃COR。 1。There is； therefore； a power of gravity tending to all the planets； for doubtless Venus； Mercury； and the rest are bodies of the same sort with Jupiter and Saturn。 And since all attraction （by Law iii。） is mutual； Jupiter will therefore gravitate towards all his own satellites； Saturn towards his； the earth towards the moon； and the sun towards all the primary planets。 〃COR。 2。The force of gravity which tends to any one planet is reciprocally as the square of the distance of places from the planet's centre。 〃COR。 3。All the planets do mutually gravitate towards one another； by Cor。 1 and 2； and hence it is that Jupiter and Saturn； when near their conjunction； by their mutual attractions sensibly disturb each other's motions。 So the sun disturbs the motions of the moon； and both sun and moon disturb our sea； as we shall hereafter explain。 〃SCHOLIUM 〃The force which retains the celestial bodies in their orbits has been hitherto called centripetal force； but it being now made plain that it can be no other than a gravitating force； we shall hereafter call it gravity。 For the cause of the centripetal force which retains the moon in its orbit will extend itself to all the planets by Rules i。； ii。； and iii。 〃PROPOSITION VI。； THEOREM VI。 〃That all bodies gravitate towards every planet； and that the weights of the bodies towards any the same planet； at equal distances from the centre of the planet； are proportional to the quantities of matter which they severally contain。
〃It has been now a long time observed by others that all sorts of heavy bodies （allowance being made for the inability of retardation which they suffer from a small power of resistance in the air） descend to the earth FROM EQUAL HEIGHTS in equal times； and that equality of times we may distinguish to a great accuracy by help of pendulums。 I tried the thing in gold； silver； lead； glass； sand； common salt； wood； water； and wheat。 I provided two wooden boxes； round and equal： I filled the one with wood； and suspended an equal weight of gold （as exactly as I could） in the centre of oscillation of the other。 The boxes hanging by eleven feet； made a couple of pendulums exactly equal in weight and figure； and equally receiving the resistance of the air。 And； placing the one by the other； I observed them to play together forward and backward； for a long time； with equal vibrations。 And therefore the quantity of matter in gold was to the quantity of matter in the wood as the action of the motive force （or vis motrix） upon all the gold to the action of the same upon all the woodthat is； as the weight of the one to the weight of the other： and the like happened in the other bodies。 By these experiments； in bodies of the same weight； I could manifestly have discovered a difference of matter less than the thousandth part of the whole； had any such been。 But； without all doubt； the nature of gravity towards the planets is the same as towards the earth。 For； should we imagine our terrestrial bodies removed to the orb of the moon； and there； together with the moon； deprived of all motion； to be let go； so as to fall together towards the earth； it is certain； from what we have demonstrated before； that； in equal times； they would describe equal spaces with the moon； and of consequence are to the moon； in quantity and matter； as their weights to its weight。 〃Moreover； since the satellites of Jupiter perform their revolutions in times which observe the sesquiplicate proportion of their distances from Jupiter's centre； their accelerative gravities towards Jupiter will be reciprocally as the square of their distances from Jupiter's centrethat is； equal； at equal distances。 And； therefore； these satellites； if supposed to fall TOWARDS JUPITER from equal heights； would describe equal spaces in equal times； in like manner as heavy bodies do on our earth。 And； by the same argument； if the circumsolar planets were supposed to be let fall at equal distances from the sun； they would； in their descent towards the sun； describe equal spaces in equal times。 But forces which equally accelerate unequal bodies must be as those bodiesthat is to say； the weights of the planets （TOWARDS THE SUN must be as their quantities of matter。 Further； that the weights of Jupiter and his satellites towards the sun are proportional to the several quantities of their matter； appears from the exceedingly regular motions of the satellites。 For if some of these bodies were more strongly attracted to the sun in proportion to their quantity of matter than others； the motions of the satellites would be disturbed by that inequality of attraction。 If at equal distances from the sun any satellite； in proportion to the quantity of its matter； did gravitate towards the sun with a force greater than Jupiter in proportion to his； according to any given proportion； suppose d to e； then the distance between the centres of the sun and of the satellite's o